Polar Coordinates Converter
Convert a point between rectangular (x, y), polar (r, θ), cylindrical and spherical coordinates. The angle comes from atan2, so every quadrant is handled correctly, and answers stay exact — 5√3 rather than 8.660254 — whenever the angle is a multiple of 30° or 45°. The point is plotted with its radius and angle, in degrees or radians. Free, no sign-up.
Whole numbers, decimals or fractions such as 3/4.
Results
5
Radius r
126.8699° ≈ 2.214297 rad
Angle θ
Quadrant II
Location
The working
r = √(x² + y²) = √((−3)² + 4²) = √25 = 5
θ = atan2(y, x) = atan2(4, −3) = 126.8699° ≈ 2.214297 rad, taken in [0°, 360°)
The reference angle is arctan(|y/x|) ≈ 53.1301°, and the point is in quadrant II, which fixes θ.
The same point has other polar forms: (r, θ) = (5, 486.8699° ≈ 8.497483 rad) after one more full turn, and (−5, 306.8699° ≈ 5.35589 rad) with a negative radius. Measured from −180° to 180°, θ = 126.8699° ≈ 2.214297 rad.
Diagram
How to Use the Polar Coordinates Converter
- 1. Choose the conversion. Rectangular to polar and back work in the plane; the other three modes handle 3D points in cylindrical or spherical form.
- 2. Pick degrees or radians. In radians you can type multiples of π as pi/6 or 2pi/3, which keeps the answer exact. For spherical coordinates, also choose the convention you are working in.
- 3. Enter the values. Whole numbers, decimals and fractions are all read exactly. The fields are labelled with the symbols of the convention you picked.
- 4. Read the result and the plot. The answer comes with its working. In the plane, the point is drawn with the radius line, the circle through it and the angle measured from the positive x-axis.
Rectangular and Polar Formulas
| Direction | Formulas |
|---|---|
| Rectangular → polar | r = √(x² + y²), θ = atan2(y, x) |
| Polar → rectangular | x = r cos θ, y = r sin θ |
| Rectangular → cylindrical | ρ = √(x² + y²), φ = atan2(y, x), z = z |
| Rectangular → spherical (ISO) | r = √(x² + y² + z²), θ = arccos(z / r), φ = atan2(y, x) |
| Spherical (ISO) → rectangular | x = r sin θ cos φ, y = r sin θ sin φ, z = r cos θ |
Why atan2 and Not arctan(y/x)
The ordinary inverse tangent returns an angle between −90° and 90°, so arctan(y/x) cannot tell (−3, 4) from (3, −4): both give y/x = −4/3, a reference angle of about 53.13°. The two-argument atan2(y, x) looks at the signs of x and y separately and returns the angle in the correct quadrant. For (−3, 4) the point is in quadrant II, so θ = 180° − 53.1301° = 126.8699°; for (−3, −4) it is in quadrant III and θ = 180° + 53.1301° = 233.1301° (or −126.8699°). The calculator reports θ from 0° to 360°, the signed equivalent from −180° to 180°, and the reference angle used.
Worked Examples
- (−3, 4) to polar. r = √(9 + 16) = 5, θ ≈ 126.8699° ≈ 2.214297 rad, in quadrant II.
- (1, 1) to polar. r = √2 and θ = 45° = π/4 exactly.
- Polar (10, 30°) to rectangular. x = 10 cos 30° = 5√3 ≈ 8.660254 and y = 10 sin 30° = 5.
- (1, 1, 1) in cylindrical and spherical form. ρ = √2 with azimuth 45°; the spherical radius is √3 and the polar angle is about 54.7356° (0.955317 rad).
- Cylindrical (2, 60°, 5) to rectangular. x = 2 cos 60° = 1, y = 2 sin 60° = √3, z = 5.
- Spherical (4, 45°, 45°) to rectangular, ISO convention. x = 4 sin 45° cos 45° = 2, y = 2 and z = 4 cos 45° = 2√2.
Two Spherical Conventions
Spherical coordinates are written (r, θ, φ) in two incompatible ways, so check which one a textbook or instructor uses. The physics and ISO 80000-2 convention takes θ as the polar angle from the positive z-axis (0° to 180°) and φ as the azimuth from the positive x-axis in the xy-plane (0° to 360°). The calculus convention used in many mathematics courses uses ρ for the radius, φ for the polar angle and θ for the azimuth. The converter has a switch for this; every symbol on the page follows the one you choose. On the z-axis the azimuth is undefined, and at the origin the angles are too, and the calculator says so instead of inventing a value.
Frequently Asked Questions
- How do you convert rectangular coordinates to polar coordinates?
- Find the distance from the origin, r = √(x² + y²), and the angle θ = atan2(y, x). Use atan2 rather than arctan(y/x) so the angle lands in the correct quadrant. The point (1, 1) becomes (√2, 45°).
- How do you convert polar coordinates to rectangular coordinates?
- Use x = r cos θ and y = r sin θ. For r = 10 and θ = 30° that gives x = 5√3 ≈ 8.66 and y = 5. A negative r simply points the opposite way, to the point r units away in the direction θ + 180°.
- Why are there many polar coordinates for the same point?
- Adding any whole number of full turns to θ returns to the same direction, so (r, θ), (r, θ + 360°) and (r, θ − 360°) are one point. A negative radius gives another form: (r, θ) equals (−r, θ + 180°). Only the origin is fully ambiguous: its r is 0 and any angle works.
- What are cylindrical coordinates?
- Cylindrical coordinates extend polar coordinates to 3D by keeping the height z: a point is (ρ, φ, z), where ρ is the distance from the z-axis and φ is the azimuth. They suit objects with an axis of symmetry, like pipes and cylinders.
- What is the difference between the physics and calculus spherical conventions?
- They swap θ and φ. In physics and ISO 80000-2, θ is the angle from the z-axis and φ is the angle around it. Many calculus texts, such as Stewart's, use φ for the angle from the z-axis and θ for the angle around it, with ρ for the radius.
- Should I use degrees or radians?
- Use whichever your course asks for; the calculator shows both. Calculus and physics usually work in radians because formulas like d/dθ sin θ = cos θ hold only then. Typing pi/6 in radian mode gives an exact answer, while a rounded decimal such as 0.5236 does not.